1. A collection of measurable sets follows a $\sigma$ algebra.

  2. All sets of measure $0$ is measurable.

My questions are:

  1. What's the significance of sets of measure $0$ in the $\sigma$ algebra of measurable sets? Are they some sort of "identity"?

  2. What's the significance of the proposition:"the translate of a measurable set is measurable." in the set of measurable set's $\sigma$ algebra? Does it somehow suggest an equivelence relation?

  3. Does the previous proposition and sets of measure $0$ carry any interesting effect?

  • $\begingroup$ Do you with "All sets of measure $0$ is measurable" refer to that if $E \subset F$ and $F$ is measurable with $m(F)=0$ (where $m$ is Lebesgue measure) then $E$ is measurable (of course with $m(E)=0$)? $\endgroup$ – md2perpe Sep 8 '18 at 18:33
  • $\begingroup$ @md2perpe Real Analyssi fourth edition by Royden and Fitzpatrick section 2.4 Theorem 11's proof the last paragraph. All sets of measure $0$ is measurable(Lebesgue), and it's not hard to prove. $\endgroup$ – user416486 Sep 8 '18 at 19:41
  • $\begingroup$ Ah, it refers to that a set $N$ of outer measure $0$ is measurable, i.e. that $m^*(A) = m^*(E \cap N) + m^*(E \cap N^c)$ for all sets $A$. $\endgroup$ – md2perpe Sep 8 '18 at 20:03
  1. Sets of measure $0$, called null-sets, are primarily significant in that functions that only differ on a null-set work exactly the same under integration. This is an equivalence relation. Such functions will later be identified. You will also see a lot of statements saying that something is valid almost everywhere meaning that it's valid except perhaps on a null-set.

  2. The proposition primarily serves to provide us with knowledge about what sets are measurable. I can't remember having seen any equivalence relation being based on translation of measurable sets.

  3. One effect that I find interesting is that there are sets that are null-sets that are so dense that every interval contains an uncountable amount of points from the set. This is however rather based on the fact that the Lebesgue measure is translation invariant, but this requires that measurability is translation invariant.


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